Answer:
0.98 seconds
Step-by-step explanation:
We assume the height of the volleyball is described by the equation for ballistic motion. We want to find the time it takes for the height to become zero.
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The general form of the equation of height for ballistic motion is ...
[tex]h(t)=-16t^2+v_0t+h_0\qquad\text{$v_0$ and $h_0$ are the initial velocity and height}[/tex]
The coefficient 16 in the equation is an approximation of 1/2g, where g is the acceleration due to gravity in ft/s². This means the units of time and distance are expected to be seconds and feet.
For the problem at hand, the initial velocity and height are 10.5 ft/s and 5 ft. Then the height equation is ...
h(t) = -16t² +10.5t +5
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Marsha has until the ball hits the ground to react to the serve. To find out how long that is, we need to solve the height equation for t when h=0. This is most easily done using the quadratic formula with ...
The solution is ...
[tex]t=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}= \dfrac{-10.5\pm\sqrt{10.5^2-4(-16)(5)}}{2(-16)}\\\\=\dfrac{10.5\pm\sqrt{430.25}}{32}=\dfrac{21\pm\sqrt{1721}}{64}[/tex]
The positive solution is ...
t ≈ 0.976327 ≈ 0.98
Marsha has about 0.98 seconds to react before the volleyball hits the ground.
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Additional comment
After about 0.33 seconds, Marsha knows she doesn't need to react at all. The serve will not clear the net. Its maximum height is about 6' 8 5/8". A women's volleyball net is 7' 4 1/8" high. Jennifer's serve velocity must be at least 12.3 ft/s for the ball to go over the net. With that upward velocity, Marsha has about 1.06 seconds to react.
